Reasons, Estimation of Parameters & IV Cont. … Error in Variable

 

Reasons, Estimation of Parameters & IV

Reasons of Errors in Variables

In classical linear regression, it is assumed that the independent and independent variables are measured without errors. In such situations, the OLS method is used to estimate the model parameters. Now, if the dependent variable is measured with error, the OLS estimates are unbiased and consistent but the variance is inflated. The prediction interval in such a situation will be wider and the model prediction will be less precise. 

The following are the main reasons for errors in variables:

i.                    Collection of Accurate Observations

In many cases the variables are well defined but the collection of precise observations can be difficult, or observations are collected from closely proxy variables. For instance, the age is generally reported in complete years or in multiples of five or and a person's education level is determined by the number of years spent in school.

ii.                  ii. The use of Qualitative Variable

In some circumstances, the variables will seem plausible and clear, but their nature may be qualitative.

Estimation of Model Parameters in case of Error in Variables

Consider the linear regression model

Y = β0 + β1 X + ϵ

Where:

y = Y + v    and x = X + u

If there are error in variables, then OLS estimates are biased and inconsistent. There is no method to give satisfactory estimates; still the following methods can be used:

1. Classical Approach

Classical approach is used to get maximum likelihood estimators of the parameters. We know that

$E \left(u\right) = 0 and V \left(u\right) = {{\sigma }_u}^2$

E (v) = 0 and V (v) = σv2

u ~ N (0, σu2) and  v ~ N (0, σv )

The joint probability function of u and v is given by;

The likelihood function is given by:

2.      Instrumental Variable Technique

The instrumental variable (IV) technique is used to estimate the cause-and-effect relationship when;

i.                    Explanatory variable(s) is correlated with the error term of the mode.

ii.                  The endogenous variable as exogenous variable in a model.

iii.                Reverse causality 

The classical assumption a linear regression model that independent variables are constant and uncorrelated with the disturbance term and there will be one-way causing phenomenon.

Consider the linear regression model

Y_ =X β _+ ϵ_

E ( X ϵ _) = 0

Y is effect and X is cause.

Now if these assumptions are violated or there is joint dependency between dependent and independent variables, then the OLS estimates are biased and inconsistent. Then to find the consistent estimates of the parameters of the above regression the technique of instrumental variable will be used.

Instrumental variable (say z) is a variable which is used as an instrument has the properties that will be correlated with regressor and will be uncorrelated with disturbance term of the model i.e.,  and do not directly affect the dependent variable.

That’s

X = α_ z + u_

The instrumental variable “z” will be uncorrelated with the error term of the model that’s cov ( z,  ϵ_) = = 0  and will be correlated with explanatory variable that’s Cov ( z X) ≠ 0 .  If z is an instrumental variable, then it must satisfy the following properties.

i.                    The instrumental variable “z” will be uncorrelated with disturbance term of the model.

ii.                    The instrumental variable “z” will be strongly correlated with dependent variable appearing as independent variable in the structural model.

iii.                  The covariance of z will be existed.

The IV estimate for regression with scalar regressor x and scalar instrument z, is defined as:

Example

Addiction to smoking is a significant risk factor for heart disease and is connected to heart attacks. The risk of heart attack is also influenced by a number of other variables, including working conditions, age, food, degree of activity, and a host of others. If we trace back a heart attack to smoking. If we regress heart attack on smoking, then it can be expressed as:

Y = α + β X + ϵ

Where:

 ϵ: captures working conditions, age, dietary habits, amount of exercise, etc.

There is a risk of smoking frequently when the work environment is stressful. As a result, the error term ( ϵ) and the explanatory variable (X: Smoking) are associated. As an instrument, we now introduce a variable tax (z) that is correlated with X but uncorrelated with the disturbance term. 

That’s 

X = γ0 + γ1 z + u

Through taxes, the rise in taxation (z) modifies smokers' smoking habits. Consequently, indirectly reduce the risk of heart attack.

 Instrumental Variable Estimator in one explanatory Variable Model.

Consider the model

Y = β0 + β1 X + ϵ

If X is correlated with  ϵ.

That’s  E ( X ϵ ) ≠ 0  

If we apply OLS method to the above model, the estimates will be biased and inconsistent. The we chose an instrumental variable “z” which is strong correlated with X and uncorrelated with the disturbance tern  ϵ and does not affect Y. It affects Y through X.

We set up X as:

X = α0 + α1 z + u

The OLS estimate of α1 is given by:

Replace independent variable by instrumental variable,

Y = β0 + β1 z + ϵ

The OLS estimate of β1 is given by:

The estimate of instrumental variable is defined as:

1.The estimate obtained through instrument variable is unbiased.

2.                    The estimate obtained through instrument variable is consistent.

Proof: We know that the instrumental variable “z” will be uncorrelated with disturbance term of the model.

Practice Question

Let the consumption (Yt) and income (Xt) be the endogenous variables and liquid assets (L) be the exogenous variable of a SEM. The relevant observations are as follow:

\Yt	Xt	L
68	91	18
70	94	19
73	96	20
74	99	25
78	102	29
83	105	33

Estimate the consumption function: Yt = β0 + β1 Xt + u

Solution:

Yt = β0 + β1 Xt + u →1

As Xt is endogenous variable which depend on liquid assets (L), so it can be written as:

Xt = α0 + α1 L + v →2

First check identification by order condition:

K = 1 (i.e., L) and k = 0 (No exogenous in eq. 1) and  m = 2

K - k = m - 1

.Now check identification by rank condition:

The consumption eq. is identified.

Note: Here we can use ILS, but so the sack of practice we using IV.

The model is express in deviated form as:

yt =  β1 xt + u 

The estimated model:

y^t =  β^1 xt 

We use liquid assets (L) as an instrumental variable.

The instrumental variable estimate is given by:

The estimated consumption equation in deviated form:

The estimated model:

Practice Question

Consider the following SEM:

Qd = β0 + β1 P + u

Qs = α0 + α1 P + α2 w + v

Determine the model parameters by ILS and IV.

Solution:

Identification by order condition:

Demand equation: K = 1, k = 0, m = 2

K - k = m - 1

The demand equation is exactly identified.

Supply equation: K = 1, k =1 and m = 2

K - k < m - 1

The supply equation is under identified.

Identification by rank condition:

The demand equation is identified and supply equation is not identified.

Now ILS method:

By IV method

Consider w is an instrument correlated with Q and uncorrelated with u.